YES
O(n)
TRS:
 {
   if(true(), X, Y) -> X,
  if(false(), X, Y) -> activate(Y),
               f(X) -> if(X, c(), n__f(true())),
               f(X) -> n__f(X),
        activate(X) -> X,
  activate(n__f(X)) -> f(X)
 }
 Natural interpretation:
  Strict:
   {
     if(true(), X, Y) -> X,
    if(false(), X, Y) -> activate(Y),
                 f(X) -> if(X, c(), n__f(true())),
                 f(X) -> n__f(X),
          activate(X) -> X,
    activate(n__f(X)) -> f(X)
   }
  Weak:
   {}
  Interpretation class: stronglylinear
  [false] = + 3
  [activate](X0) = + 1*X0 + 3
  [f](X0) = + 1*X0 + 3
  [true] = + 0
  [n__f](X0) = + 1*X0 + 1
  [c] = + 0
  [if](X2, X1, X0) = + 1*X0 + 1*X1 + 1*X2 + 1
  
  Qed